Problem: After a special medicine is introduced into a petri dish containing a bacterial culture, the number of bacteria remaining in the dish decreases rapidly. The population loses $\dfrac 14$ of its size every $44$ seconds. The number of remaining bacteria can be modeled by a function, $N$, which depends on the amount of time, $t$ (in seconds). Before the medicine was introduced, there were $11{,}880$ bacteria in the Petri dish. Write a function that models the number of remaining bacteria $t$ seconds since the medicine was introduced. $N(t) = $
Solution: The strategy We can model the situation with an exponential function of the general form A ⋅ B f ( t ) A\cdot B\^{ f(t)}, where $A$ is the initial quantity, $B$ is a factor by which the quantity is multiplied over constant time intervals, and $f(t)$ is an expression in terms of $t$ that determines those time intervals. Let's use the given information to determine $A$, $B$, and $f(t)$. Understanding what's given We are given that the initial number of bacteria is $11{,}880$, and the bacterial culture loses $\dfrac 14$ of its size every $44$ seconds. Note that losing $\dfrac 14$ is the same as being multiplied by $\dfrac 34$. [Why?] This means that the initial quantity is $A=11{,}880$ and the factor is $B=\dfrac 34$. We need to find $f(t)$ based on the fact that the quantity is multiplied by $\dfrac 34$ every $44$ seconds. Finding the expression in the exponent We know that the number of bacteria is multiplied by $\dfrac 34$ every $44$ seconds. This means that each time $t$ increases by $44$, $f(t)$ increases by $1$. Therefore, $f(t)$ is a linear function whose slope is $\dfrac{1}{44}$. When the initial measurement is made, the number of bacteria hasn't changed. So $N(0) = 11{,}880$, which means that $f(0)=0$. [Why?] Therefore, $f(t)$ must be $\dfrac{t}{44}$. Summary We found that the following function models the number of bacteria $t$ seconds since the medicine was introduced. N ( t ) = 11,880 ⋅ ( 3 4 ) t 44 N(t)=11{,}880\cdot \left(\dfrac 34 \right)\^{ \frac{t}{44}}